What is the difference between the slope of a tangent and instantaneous rate of change of #f(x)#?
1 Answer
There is no difference in the mathematical expressions for these things. The difference is interpretation (and setting).
The slope of a tangent line is a geometrical idea. It is tied to curves in a coordinate plane.
The instantaneous rate of change is a relationship between two variable quantities, one depending on the other.
The calculations or the algebra, if you like, are the same for both.
Consider:
In terms of the symbols we write mathematically, there is no difference between the slope of a secant line, (geometry) the average rate of change (related variable quantities) and the difference quotient (an algebraic expression). But the mathematics has different meanings in different settings.
In a similar way (in the same way?):
